We now discuss another way of showing that the nodal abscissas in Gauss-Legendre quadra- ture are given by roots of Legendre polynomials. Take f(x) to be a polynomial of degree up to 2n 1; divide f(x) with Pn(x), the Legendre polynomial of degree n: f(x) = Qn-1(2)Pn(x) + Rn-1(x) (7.243) where the quotient and remainder are polynomials of degree up to n-1. Integrate both sides from 1 to +1 and justify why one of these integrals vanishes. Express the non-vanishing integral as a sum over weights and function values, as in our defining Eq. (7.7). Now take the It's to be the roots of Pn(x) and re-express the sum using Eq. (7.243) evaluated at the xk's. f(x)dx = {cif(x) (7.7) n-1 S i=0 We now discuss another way of showing that the nodal abscissas in Gauss-Legendre quadra- ture are given by roots of Legendre polynomials. Take f(x) to be a polynomial of degree up to 2n 1; divide f(x) with Pn(x), the Legendre polynomial of degree n: f(x) = Qn-1(2)Pn(x) + Rn-1(x) (7.243) where the quotient and remainder are polynomials of degree up to n-1. Integrate both sides from 1 to +1 and justify why one of these integrals vanishes. Express the non-vanishing integral as a sum over weights and function values, as in our defining Eq. (7.7). Now take the It's to be the roots of Pn(x) and re-express the sum using Eq. (7.243) evaluated at the xk's. f(x)dx = {cif(x) (7.7) n-1 S i=0